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By Chambadal L., Ovaert J.-L.

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Let X be any normal irreducible scheme and D be a Weil Q-divisor D with qi the Seifert divisor D − D = ei Ei . The computations from above show that ∞ the sheaf of graded algebras ⊕i=0 OX (iD), at a geometric generic point η¯i of Ei looks like the OX,¯ηi (U, Z]µe , where the action is given as above by the numbers q i , ei . 4 Cylinder constructions Let L be an invertible sheaf on X. Consider the projective vector bundle P(L ⊕ OX ). It is convenient to identify S • (L⊕OX ) with the graded algebra S • L[t] locally isomorphic to the graded polynomial algebra O(U )[tU , t].

15) Proof. kS/T (E1 +E2 ) = −(E1 +E2 )2 −2χ(OE1 +E2 ) = −E12 −E22 −2E1 ·E2 −2χ(OE1 +E2 ) = kS/T (E1 ) + kS/T (E2 ) = −E12 − E22 − 2χ(OE1 ) − 2χ(OE2 ). This proves the assertion. 3. Let dim T > 0 and D be a divisor with support in connected fibre f −1 (t), where κ(t) is algebraically closed. Assume that D is reducible if dim T = 1. Suppose that kS/T (E) = 0 for each irreducible component of D. Then the intersection matrix of D is a Cartan matrix. Proof. Let Ei be an irreducible component of D. Since κ(t) is algebraically closed and Ei is a projective connected reduced scheme, we have H 0 (Ei , OEi ) ∼ = k.

For example, suppose A = S • E for some locally free sheaf E, a surjective homomorphism q ∗ A → S • L is defined by a surjective homomorphism of OX -modules q ∗ E → L. 12 from Chapter II of [Hartshorne]). Here, for any locally free sheaf E on S we set P(E) = Proj S • E. It is called the projective bundle associated to E. e. by a collection of r + 1 sections of L not vanishing simultaneously at any point of X. 2 Ample invertible sheaves Let X be a proper scheme over a noetherian ring R and L be an invertible sheaf on X.

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